Question

How do you factor `3a^2 + 14a − 24`?

Answer 1

`(3a - 4)(a + 6)`

Explanation:

We know there must be a `3a` and `a` in the brackets because they are the only factors of `3`.

You then use trial and error to find a the other numbers which multiply together to make `24` and add together (multiplied by their opposite coefficients of `a`) to make `14`, which you find are `6` and `-4`.

Answer 2

(3a - 4)(a + 6)

Explanation:

Use the systematic, non-guessing new AC Method (Socratic Search).
`y = 3a^2 + 14a - 24 =` 3(a + p)(a + q)
Converted trinomial `y' = a^2 + 14a - 72 =` (a + p')(a + q')
p' and q' have opposite signs because ac < 0.
Factor pairs of (ac = -72) --> (-3,24) (-4, 18). This sum is 14 = b. Then,
p' = -4 and q' = 18.
Back to trinomial y, `p = (p')/a = - 4/3` and `q = (q')/a = 18/3 = 6`
Factored form: `y = 3(a - 4/3)(a + 6) = (3a - 4)(a + 6)`